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Chapter 1: Problem 6

Multiply. $$3 \cdot(-11)$$

Short Answer

Expert verified
The short answer for the given multiplication problem is: \(3 \cdot(-11) = -33\).

Step by step solution

01

Identify the sign of the result

We are given a positive integer (3) and a negative integer (-11) to multiply. When multiplying a positive integer and a negative integer, the result will always be negative.
02

Multiply the absolute values of the numbers

We need to multiply the absolute values of the given numbers, ignoring their signs. Absolute value of 3 is 3 and absolute value of -11 is 11. \(3 \times 11 = 33\)
03

Apply the sign to the result

Since the multiplication result is negative (as identified in step 1), the answer is -33. So, \(3 \cdot(-11) = -33\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Positive and Negative Integers
Understanding integers is crucial for mastering arithmetic operations like multiplication. Integers include all whole numbers and their negative counterparts. Examples are -3, -2, -1, 0, 1, 2, 3, etc.

When dealing with integers, it's important to know their signs:
  • Positive integers are those greater than zero, like 1, 2, 3, etc.
  • Negative integers are less than zero and have a minus sign, like -1, -2, -3, etc.
Signs play a vital role in determining the result of mathematical operations. In multiplication, these signs help dictate whether the result will be positive or negative. This concept becomes especially apparent in the next sections dedicated to multiplication rules and the absolute value.
Absolute Value
The absolute value of a number refers to its distance from zero on the number line, without considering its sign. For example, the absolute value of both -11 and 11 is 11, because they are each 11 units from zero.

Understanding absolute value is essential when multiplying integers, as it lets us focus on the magnitude of the numbers without worrying about their signs at first.

To find the absolute value of a number:
  • If the number is positive, its absolute value is the number itself. For instance, the absolute value of 3 is 3.
  • If the number is negative, its absolute value is the same number without the negative sign. For example, the absolute value of -11 is 11.
Once we've found the absolute values, we can multiply these numbers and apply the sign afterward, which is guided by the multiplication rules you'll read about next.
Multiplication Rules
When multiplying integers, a crucial aspect is knowing the rules that govern the signs of the resultant product. These rules allow us to determine whether the final answer is positive or negative:

  • Multiplying two positive integers results in a positive product. For example, 2 multiplied by 3 is 6.
  • Multiplying two negative integers also results in a positive product, since the negatives "cancel out." An example is -2 multiplied by -3 equals 6.
  • Multiplying a positive integer by a negative one gives a product that is negative. Our example, 3 multiplied by -11, yields -33.
These straightforward guidelines help students correctly find the sign of their product and understand the foundational rules that govern integer multiplication. By using these rules along with the absolute values explored earlier, students can simplify and solve multiplication problems with confidence and clarity.

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